Turbulence in Climate Studies

Turbulence in Climate Studies

Ravi S Nanjundiah

Centre For Atmospheric and Oceanic Sciences Indian Institute of Science

Bangalore-560012 email:[email protected]

Outline

Climate: An Interactive System

Climate Interactions Rotation

Zonal Mean Circulation Assymetry Modelling Climate The Equations Equations Simplifications Resolution Numerical Techniques Turbulence Turbulence in Atmospehere Clouds Cumulus Parameterization Conditional Moist Static Instability Stable Condensation

Kuo-Anthes Parameterization

kuo scheme

Arakawa Schubert Scheme

Overview of Arakawa-Schubert Scheme

The Cloud Work Function

Buoyancy and Cloud Work Function Factors affecting Cloud Work Function

Algorithm

algoritm

Climate: An Interactive System

source: Internet

• Climate is influenced by many factors: external and internal

• It is a coupled system in which atmosphere,ocean,land surface interact with each other

• Atmosphere drives ocean currents through winds

• Ocean impacts atmosphere through Sea Surface Temperature and moisture fluxes

• Atmosphere interacts with

land-surface: transfer of sensible heat and frictional effects

• Precipitation leads to percolation into the deep soil and also to runoff which could flows into the sea through rivers • Rivers can also influence oceans:

fresh water discharge can change the salinity and hence stratification: such effects play a role in modulating Monsoons over Bay of Bengal.

Climate . . .

• Phase change between vapour, water and ice is important: latent heat release changes temperature and hence changes atmospheric circulation

• Atmospheric circulation through advection plays a major role in transporting moisture leads to a coupling between

thermodynamics and fluid dynamics • Multiple scales exist and interact

continously with each other

• Water evaporates from ocean/land surface at extremely small scales

• Eddies carry them upward away from the surface

• Advection carries it over long distances • Condensation occurs on cloud scales∼few km

• The heating in clouds induces large scale flow – energy transfer from smaller to larger scales.

• Turbulence plays a major role in the boundary (where evaporation occurs) and in clouds (where condensation occurs)

Earth’s Rotation

source: wikipedia.org

• Earth’s rotation also plays an important in atmospheric/oceanic circulation

• The effect of rotation is different at different latitudes

• Near the pole where the axis of rotation is close to the local vertical, the effects are more pronounced – geostrophic balance

• As we move to the tropics the rotation vector moves more and more towards the horizontal plane (meridional-zonal plane)

• Geostrophic balance i.e. balance between Coriolis and horizontal pressure gradient is much weaker in tropics

• Exact role of rotation on tropical circulation (especially near equator) is still a matter of debate

• Heating (latent heat release in clouds) is more pronounced in tropics as compared to extra-tropics

• Turbulence and Non-linear interactions (within atmosphere and with ocean/land) play an important role in tropics

The Time Averaged Circulation

• Unequal heating between equator

and poles causes an equator-to-pole heat transfer

• Near equator air rises and sinks

near 300_{– the Hadley Cell}

• It also gets deflected eastward

due to earth’s rotation (westerlies)

• However as one moves

polewards, rotation becomes dominant

• Atmosphere is also more

baroclinic (steeper horizontal

temperature gradients)→leads

to eddy formations→baroclinic

instability

• Though the mid-latitude circulation is shown as the Ferrel Cell, it is

actually large-scale baroclinic eddies and fronts which transfer energy towards the poles

The Walker Circulation

Source: Internet

• There is rising motion off the Indonesian coast (Western Pacific) • Descending motion off the American

coast

• This circulation is related to Sea Surface Temperature (SST) gradients in the East-West direction along the equator

• Water is warmer along the Indonesian (West Pacific coast) and cooler off the American coast

• Warmer SST regions are more amiable for moist convection

• The relationship between SST and moist convection (rainfall) is non-linear

• Till about 26o_{C rainfall is very low and increases rapidly} between 26o_{C -27.5}o_C

• Beyond 27.5o_{- a high SST is necessary but not} sufficient condition for moist convection

Ocean-Atmospheric Coupling

• Looking at the surface wind pattern we notice that the winds are generally westward (easterlies, the trade winds) • The easterlies cause upwelling at the

eastern end of a basin and along the equator - causes the thermocline (the interface across which there is cold water below and warmer water above) to be shallow.

• Shallower thermocline allows cold water to come up easily at the Eastern (American) end – stays cold – inhibits moist convection

• West (Indonesia) stays warm – supports convection

• At the western end there is a ’piling up of water’ due to westward wind – the thermocline is deep

• This coupled interaction leads to the Walker circulation

• The collapse of this circulation due to changes in coupled interaction causes El-Nino

The El-Nino

• In some years the easterly trades weaken

• Many theories about the cause of this: one of the more prevalent ones is

about westerly wind bursts related to rainfall activity on intraseasonal scale (the Madden-Julian Oscillation)

• The weakening of easterlies reduces upwelling and the thermocline

deepens over Central and Eastern Pacific

• These two regions now become warm and can also sustain convection

• Moist convection shifts from Western Pacific to Central and Eastern

Variations from the Mean Picture

• Time mean picture is not quite representative

• On a day-to-day basis there are large deviations from this mean picture (any animation of satellite pictures will show this!)

• There are large assymetries in rainfall e.g. Saudi Arabia (very dry) and North Eastern India (one of the wettest spots on earth) lie at the same latitude. • Mountains complicate the picture

The Components

• We need to model

• Processes in the atmosphere

• Processes in the ocean

• Processes that govern land surface parameters

Modelling the Climate

• We have seperate models for

atmosphere, ocean, land surface and sea-ice which couple to each other

• Most often these components are

stand-alone models which can be run by themselves with

The Model Equations

• We use equations for conservation of mass, momentum, energy and equation of state

• In addition if we advect additional tracers (such as green house gases, aerosol pollutants etc) we include conservation equations for them

• The mass conservation equation 1

ρ Dρ

Dt + ∇ ·U~ =0 • The momentum conservation equation: D_Dt~U =−2Ω~×_U~−1

ρ∇p+~g+Fr • Energy Conservation Equation

Cv

DT

Dt +p

Dα

Dt =J

• The J term includes radiative heating, effect of cloud heating, transfer of sensible heat from the surface

• Radiative heating includes incoming solar radiation (mostly in visible part of spectrum) and out going terrestrial radiation (essentially in the infra-red) • Computationally this could take upto 40% of the overall computing time • Longwave emissivities and absorptivities computational can take 2-3 times the

computational taken by rest of the atmospheric model • Equation of state :P=ρRT

Simplifications

• Generally while looking at the global scale, we find that the aspect ratio (vertical/horizontal) is very small.

• Vertical scale is small about 10 km

• Horizontal scale could rangle from 100 km - 10000 km (depending on the phenomena being studied

• When looking at larger scales we use the ‘thin-shell’ approximation

• Radius r = a +z, where z is vertical distance from mean sea level, ‘a’ the nominal radius of earth’s surface

• All radial distances are replaced with ‘a’ and radial derivatives by derivative wrt ‘z’

• While considering large scale flows, typical horizontal velocity is about 10m−1_s

• Vertical velocity is 1-10cm−1_s.

• To a first approximation we assume that vertical component of momentum equation can be replaced by hydrostatic balance∂_∂p_z =−ρg

• This removes vertical acoustic modes

• Also removes the effect of Coriolis acceleration in the vertical (and hence for consistent energetics needs to be dropped from zonal momentum equation)

• Curvature terms are also sometimes dropped

• Flow looks quasi-two-dimensional (though clouds and boundary layer turbulence are more prominent in the vertical)

• Now vertical velocity needs to be calculated indirectly through continuity equation

Typical Resolutions used in Climate Models

• Climate models (using coupled Ocean-Atmospheric Models) need to be

integrated for 100+ years to study climate change

• Coarse resolution∼100km on a global scale

• Seasonal Forecasts (also with coupled models) are run at similar

resolutions (typical duration is 9 months)

• Medium Range forecasts (upto 15days) are conducted with atmospheric

global models – resolution is about 30 kms

• Short range forecasts for short periods upto 3 days – with regional

models, resolution – 5-10 km

• Very high resolution location specific forecasts could have resolution of

Numerical Techniques Used in Climate Models

• For weather prediction on local scale (high resolution, short range) Finite

Difference methods are preferred (these models are usually non-hydrostatic)

• For Global medium range weather forecasts spectral methods (due to

their higher accuracy) are preferred. Both NCEP & ECMWF use spectral models

• Climate models, also advect additional tracers (such as GHGs, aerosols)

• The concentration of these could vary substantially over space – not

smooth functions

• Sharp variations are resolved poorly (Gibbs oscillations) by spectral

models hence not widely used

• Climate Models use finite volume/Semi-Lagrangian techniques as these

techniques are also highly scalable (also large time-steps possible)

• Some models such as NICAM have used icosahedral technique for

obtaining higher resolution and scalability

Turbulence

• We have seen that most models have horizontal resolution of about

10-100 km

• Vertical resolution is at best∼100 m. Most models have higher

resolution in the boundary layer and near tropopause but coarser everywhere.

• Surface fluxes of momentum, heat and moisture play a significant role in

determing atmospheric circulation

• These processes occur at smaller scales which cannot be resolved (or

likely to be resolved in the near future)

• Hence the effects of boundary layer are parameterized

• The common approaches are

1. Mixed Layer Models

2. ‘Local’ closures based on eddy diffusivity

3. ‘Nonlocal’ closures

• We are nowhere close to LES (let alone DNS!) for BL in a climate model

Impact of Boundary Layer Parameterization On Monsoon Rainfall

• Simulations are with WRF at 18 km resolution

• Rainfall during July is sensitive to boundary layer scheme

• Seems to improve using the Mellor-Yamada Janjic scheme (which uses a higher order closure)

What do Clouds Do?

• Clouds influence climate system

by:

1. Coupling Dynamical and Hydorological processes through latent heat of condensation, redistributing sensible & latent heat and momentum

2. Coupling radiative and dynamical-hydrological processio in the atmosphere through reflection, absorption and emission of radiation

3. Influencing hydrological process in the ground through

precipitation

4. Influencing the coupling between atmosphere and oceans (or ground) through modification of PBL and radiative processes

• All the interactions are two-way: Clouds influence the large-scale and

vice-versa

• These interactions need to be included in addressing the cumulus

Schematic of A Cloud

• Updrafts, Downdrafts, Rainfall, Re-evaporation of rainfall

• Low level Convergence of moisture, Outflow at the top, lateral

What Should Cumulus Parameterization Calculate?

• The cumulus parameterization is ’formulation of statistical effects of

moist convection to obtain a closed system for predicting weather and climate’

• Arakawa (2004) categorizes the computations in a cumulus

parameterization into ‘Classical and Non-Classical Objectives’

• Classical Objectives include:

1. Vertically integrated cumulus heating: This is the most basic objective. It is also related to the total convective precipitation falling to the ground

2. Vertical distribution of Cumulus heating and Drying of the atmosphere

• Non-Classical Objectives:

1. Mass Transfer by Cumulus convection: this would involve moving of lower level air to higher levels (and in the process advection of tracers), entrainment and detrainment and mixing with environmental air.

2. Generation of Liquid and Ice Phases of Water

3. Interaction with Planetary Boundary Layer : all the low level air that causes clouds is from the Planetary Boundary Layer. Diurnal cycle and variation of PBL with it could have an significant impact on formation of clouds

4. Interaction with radiation: clouds influence radation which in turn influences formation of clouds

5. Transport of Momentum: Momentum is transported in cumulus convection but Arakawa considers this the most difficult part of parameterization

Cumulus Parameterization

• In cumulus parameterization we do not model every individual cloud (we

cannot resolve every cloud)

• However the effect of a statistical ensemble of clouds on the

environment is incorporated

• This effect needs to be incorporated in terms of the large-scale variables

(we call this the closure problem – similar to the one in PBL)

• Clouds may not be present at every grid point or at all levels at a single

grid point

• We use a set of necessary and sufficient conditions to check for the

possible occurrence of clouds

• These conditions could be

• Presence of large-scale low level moisture convergence

Conditional Moist Static Instability

• Instability implies buoyancy of a parcel of air vis-a-vis environment

• If unstable the parcel will move up further in the direction of displacement

• At the surface a moist parcel is not generally saturated

• However on raising above the lifting condensation the moisture

condenses, heats the parcel and makes it more bouyant− − −>

making the parcel unstable

• This is unknown as conditional moist static instability

• We now look at parameterizations that use either moist static stability or

large-scale forcing to incorporate the effect of clouds

• Sometimes precipitation can also occur just by supersaturation with little

Stable Condensation

• This occurs when the lapse rate is stable but the atmosphere becomes

saturated

• We can represent this asγ < γmandq>qs

• The excess moisture is condensed, the latent heat released warms the

air

• We could consider it the reverse of the wet-bulb process.

• We need to change the temperature and specific humidity in a layer such

that it is stable and saturated (or sometimes slightly less than saturated)

• We can write this as

−Lδq=cpδT

and

−δq=q−qs(T+δT,p)

• We get an implicit system of equations in q and T

• This can be solved iteratively

• Sometimes we take into consideration re-evaporation of rainfall i.e.δqat

Kuo-Anthes Parameterization Scheme

• Used essentially for hurricane models.

• This scheme essentially depends on the occurrence of large-scale

convergence to model cumulus convection.

The vertically integrated moisture convergenceMt is sum of

convergence and surface evaporation and is given by:

Mt=− 1 g Z Po 0 ∇ ·(qV dp~ +Evap

• Inside the cloud it is assumed that the temperature profile follows a moist

adiabat θe=T p∗ (p−es) κ e Lv qs Cp T

• It is assumed that this vapour is used to make cloud column with

temperature and moisture(Tc,qc)from environmental air at(T,q)

• Part of this vapour will condense, raising the temperature fromTetoTc

W1= Cp L Zpb pt (Tc−Te) dp g

Kuo-Anthes . . .

• Reminder will increase the specific humidity of the cloud column fromqe

toqc W2= Z pb pt (qc−q) dp g

• Hereptis pressure at the top of a cloud andpbpressure at the bottom.

For deep clouds we can writepb≈poandpt ≈0

• Total vapour needed to create cloud over unit area is

W=W1+W2

• We can write rate of cloud production C is assumed to be proportional to

the convergence plus evaportion divided by amount of vapour necessary to produce the model cloud:

C=Mt W = −1 g RPo 0 ∇ ·(qV~)dp+Evap Cp L Rp_b pt (Tc−Te) dp g + Rp_b pt (qc−q) dp g

• This in other words this is ratio of available moisture to reqd moisture

Kuo-Anthes . . .

• Rate of latent heatingQis given by

Q(p) = Ccp[Tc(p)−T(p)]ifT <TcandMt >0 = 0ifT ≥TcorMt ≤0

• Now assume that the large-scale forecast is made for a time step∆t, the

temperature without cumulus effect beT∗

• Let temperature after cumulus parameterization beT(t+ ∆t)given by

T(t+ ∆t) =T∗+C∆t(Tc−T∗)

• Corresponding equation forqis gievn by

q(t+ ∆t) =q∗+C∆t(qc−q∗) • We can write rate of precipitation asR(p) =Ccp

L(Tc−T ∗

) • Total rainfall at the surface would beR=Rpo

o C cp

L(Tc−T ∗

Kuo-Anthes . . .

• Not all moisture that converges precipitates out. Some part moistens the atmosphere, the rest precipitates out.

• This partitioning is done on the basis of a parameterb

b=1− q

qs where¯q=R1

0 qdσandq¯s=

R1

0 qsdσ. Here,qsis the saturated specific humidity.

• The total moistening isbMt

• Actual clouding will be given byC=(1−b)Mt

W • Heating will be given as before by

T(t+ ∆t) =T∗+C∆t(Tc−T∗) • We can write rate of precipitation asR(p) =Cc_Lp(Tc−T∗) • Total rainfall at the surface would beR=Rpo

o C cp

L(Tc−T ∗_)dp • Corresponding equation forqis given by

q(t+ ∆t) =q∗+C∆t(qc−q∗) • Kuo-Scheme generally overestimates rainfall over oceans • It assumes that there is one type of cloud

Arakawa-Schubert Scheme

• This scheme is based on theory of interaction between a cumulus

ensemble (a set of clouds) and the large-scale environment

• The scheme operates on the presence of static instability rather than on

occurrence of large-scale low-level moisture convergence scheme

• Most modern day schemes such as the Simplified Arakwa-Schubert

(SAS) and Relaxed Arakawa Schubert (RAS), the Zhang-McFarlane mass flux scheme are off-shoots of the Arakawa Schubert Scheme

Overview of Arakawa Schubert Scheme

• In the Kuo-Anthes, a single cloud type, here a set of clouds each set

(sub-ensemble) with a distinct cloud top and an entrainment rate

• Works on the principal that large-scale environment becomes unstable

due to various processes And

• Occurrence of clouds tend to remove this instability

• Instability determines cloud base mass flux

Which is related to

• A concept calledCloud Work Function

• Works on the principle of quasi-equilibrium i.e. at the end of the

application of the cumulus scheme, the instability is completely removed

• This scheme considers three regions for energy and moisture budget

equations

1. Lower most layer : sub-cloud layer or mixed layer

2. Clouds

Buoyancy and Cloud Work Function

• Cloud Work Function represents the rate of Kinetic Energy generation by

buoyancy force

• Buoyancy Force depends on

• properties of cloud

• properties of environment

• The cloud work function (essentially Convective Available Potential

Energy) is given by:

A(λ) = Z Zt Zb gCp Te(z) η(z, λ)[Svc(z, λ)−Sve(z)]dz

Cloud Work Function . . .

• Cloud work function is an integral of bouyancy force that governs the K.E

generation in a ensemble

• Bouyancy affected by entrainment rateλ

• If the cloud ensemble does not dissipate within the disspative time scale

Factors Affecting Cloud Work Function

• Clouds form due to instability of the environment – static control

• When clouds form they transport heat and moisture flux upwards

• Reduces the instabilityA(λ)– dynamic feedback

• We can divide rate of change of A into two parts

dA(λ) dt = dA(λ) dt c + dA(λ) dt LS

• LSrepresents environment,crepresents cloud

• We writehdA_dt(λ)i cin a symbolic form as dA(λ) dt c = Z λmax 0 K(λ, λ0)mb(λ)dλ0

• This essentially represents how clouds influence each other

• K(λ, λ‘)(usually <0) represents how clouds influence each other. Presence of other clouds will reduce the buoyancy in a cloud.

The Algorithm

1. First do a large scale forecast without cloud effects. This will non-zero

cloud work functionA(λ)

2. Make cumulus adjustment to large scale parameters so that cloud work

function is near zero.

3. Difference between above two givedA_dt∆tuse this forF(λ) 4. Calculatemb(λ)fromRλmax

o K(λ, λ‘) +F(λ)≈0 5. Calculate changes inT¯ and¯qusing

∂¯s ∂t =−V¯· ∇¯s−ω¯ ∂¯s ∂p +g ∂ ∂pFs−Ll+LR+QR ∂q¯ ∂t =− ¯ V· ∇q¯−ω¯∂q¯ ∂p+g ∂ ∂pFq+l−R

6. Total rainfall is integral of R over the entire column where R(p) is given

by:

R(p) = Z λD(p)

0

Algorithm . . .

• Arakawa-Schubert find that usingλfor integration is cumbersome.

• Since for a given entrainment rate and environment conditions, the

height of detrainment is fixed, the replaceλbyZD(p)– this is the height

Sensitivity to Cumulus Parameterization

• Simulations are very sensitive to prescription of cumulus

parameterization

Simulations With A Very High Resolution Model

source: Oouchi et al, 2004, GRL L11815

• A 7 km icosahedral

non-hydrostatic model (NICAM) was used for a season’s simulation

• This does not use any

cumulus parameterization scheme. Tries to explicitly model the clouds

• Higher resolution appears

to help though some problems remain

Concluding Remarks

• Climate is a result of multi-scale, multi-component system interacting

continously

• Turbulence plays an important role in boundary layers and clouds

• Particularly in tropics, clouds and turbulence could be more important

• These appear to be the weakest links in climate system modelling

• Long way before we can think of LES in a climate system model

• However results of LES and DNS could be used for improving the

boundary layer parameterizations

• Additionally in the tropics, free convection at low wind speeds could have

a significant impact on turbulent fluxes (as shown by Rao and Narasimha,2006) – this is missing in almost all models.

• Similar exercises with clouds and cloud-like flows could help in improving

cumulus parameterization

• Perhaps a scheme that addresses both boundary layer and cloud

convection simultaneously would improve simulations (‘Unified Parameterization’)